The motivation for studying (REF ) comes from two distinct sources. The first is the
stabilizing phenomenon observed in physical experiments involving electrically conducting fluids.
The experiments exhibit a remarkable phenomenon: a background magnetic field actually stabilizes
and damps turbulent MHD fluids (see, e.g., [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}). We intend to establish this phenomenon as a mathematically
rigorous fact on (REF ). The second is to initiate new strategies and develop innovative tools for stability and large-time behavior problems on anisotropic models.
